When calculating the critical speed of a shaft supported by air bearings, several key factors must be considered to ensure accuracy and reliability. Understanding these factors is crucial as they directly impact the performance and safety of the rotating machinery. Critical speed calculations help in predicting the point at which the shaft begins to resonate, leading to potential damage or failure.

** Air bearings, with their high stiffness and minimal friction, provide significant advantages in high-speed applications.** They reduce the wear and tear typically associated with traditional bearings and offer smoother and more stable operation. However, it is important to account for both the bearing properties and the shaft characteristics in the calculations.

**Customer Considerations**

**1. Shaft Properties**

– Material Properties: Young’s modulus (E) and density of the shaft material.

– Mass Distribution: Even or uneven mass distribution along the shaft.

**2. Air Bearing Properties**

– Damping: Air bearings provide damping characteristics, which can affect vibration behavior.

**3. Load Conditions**

– Weight of the Shaft: The static deflection (δ) is influenced by the weight of the shaft.

– External Loads: Any additional loads applied to the shaft during operation.

**4. Boundary Conditions**

– Support Locations: The position of the air bearings along the shaft.

**5. Operating Environment**

– Pressure: The air pressure supplied to the bearings.

**6. Dynamic Effects**

– Rotational Speed: The speed at which the shaft is intended to operate.

– Resonance: Ensuring that operating speeds avoid critical speeds to prevent resonance.

**7. Measurement and Validation**

– Deflection Measurement: Accurately measuring the static deflection (δ) of the shaft.

Calculation Steps

Calculation Steps

**1. Measure or Calculate Static Deflection (δ)**

Use the formula for deflection of a simply supported beam if applicable:

– W is the weight of the shaft (N).

– E is Young’s modulus (Pa).

**2. Calculate Effective Stiffness (K_eff)**

Use the formula to calculate the effective stiffness: K_eff = 1 / (1/K_shaft + 1/K_bearing)

**3. Calculate Static Deflection (δ)**

**4. Calculate Critical Speed**

Critical Speed (in RPM) = (1 / 2π) * √(g / δ) * 60

δ is the static deflection (m).

**Damping and Friction Considerations**

Air bearings introduce minimal friction, but any residual friction can introduce damping into the system. Damping reduces the amplitude of vibrations at and near the critical speed, making the system more stable.

**5. Calculate Damping Ratio (ζ)**

Calculate the mass (m) of the shaft.

6. Calculate Damped Natural Frequency (ω_d)

7. Convert to RPM

Damped Critical Speed (in RPM) = ω_d / 2π × 60

Example Calculation

Example Calculation

**Calculate Static Deflection (δ)**

**Calculate Critical Speed**

**Calculate Damping Ratio (ζ)**

Damping Ratio (ζ): ζ = 100 / (2√(10 × 0.67 × 10⁶)) ≈ 0.0193

**Calculate Damped Natural Frequency (ω_d)**

**Convert to RPM**